Question

x $\ge$ sin(x) for $\forall$ x $\ge$ 0

• A. True
• B. False

Answer 1

The correct option is A

True

There are multiple ways to solve this question. Here, we will look at how it can be solved using the concept of monotonicity.

Let’s define a function f(x) such that f(x) = x - sin(x)

f(x) = x - sin(x)

By simply putting x = 0 we can see that f(0) = 0

Now, if somehow we could prove that f(x) is a monotonically increasing function in the interval (0, $\infty$) then we can say that f(x) $\ge$ 0 $\forall x\epsilon$ (0, $\infty$)

Or in other words we can say that x $\ge$ sin(x)

Now let’s see whether the function is monotonically increasing or not in the interval of (0, $\infty$)

f(x) = x - sin(x)

f’(x) = 1 - cos(x)

We know that cos function varies from [-1, 1] so the minimum value f’(x) can attain is zero.

So, f’(x) $\ge$ 0

And we know that if f’(x) $\ge$ 0 then f(x) is monotonically increasing.

Since the function f(x) is monotonically increasing after x =0 and at x =0 the value of the function is 0, we can say that after x=0 f(x) will be giving only nonnegative values.

Or x - sin(x) $\ge$ 0

Or x $\ge$ sin(x).

So the given statement is correct